A120600 G.f. satisfies: 7*A(x) = 6 + x + A(x)^6, starting with [1,1,15].

1, 1, 15, 470, 18390, 805806, 37828981, 1860433080, 94614523740, 4935081398830, 262560448214031, 14193030016877406, 777315341935068820, 43039297954660894560, 2405249540028525971070, 135492504636185052358656, 7685561884110284691089331, 438601892971571496262327410, 25164821367929258310475059330

Offset

0,3

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Links

Robert Israel, Table of n, a(n) for n = 0..557

Formula

G.f.: A(x) = 1 + Series_Reversion(1+7*x - (1+x)^6). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (6+x)^(5*n+1)/7^(6*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ (-6 + 5*(7/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 7^(2/5) * n^(3/2) * sqrt(5*Pi)). - Vaclav Kotesovec, Nov 28 2017

D-finite with recurrence: -72*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 864*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 349920*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 1399680*(n + 3)*(n + 2)*(n + 1)*(72*n^2 + 432*n + 641)*a(n + 3) - 151165440*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 4856069*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5). - Robert Israel, Mar 23 2026

Example

A(x) = 1 + x + 15*x^2 + 470*x^3 + 18390*x^4 + 805806*x^5 +...
A(x)^6 = 1 + 6*x + 105*x^2 + 3290*x^3 + 128730*x^4 + 5640642*x^5 +...

Maple

f:= gfun:-rectoproc({-72*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 864*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 349920*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 1399680*(n + 3)*(n + 2)*(n + 1)*(72*n^2 + 432*n + 641)*a(n + 3) - 151165440*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 4856069*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5), a(0) = 1, a(1) = 1, a(2) = 15, a(3) = 470, a(4) = 18390}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 23 2026

Mathematica

CoefficientList[1 + InverseSeries[Series[1+7*x - (1+x)^6, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+x+15*x^2+x*O(x^n)); for(i=0, n, A=A-7*A+6+x+A^6); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120599, A120601 - A120607.

Sequence in context: A306609 A272905 A005815 * A377498 A377497 A279922

Adjacent sequences: A120597 A120598 A120599 * A120601 A120602 A120603

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Extensions

More terms from Robert Israel, Mar 23 2026

Status

approved